CHAPTER 13
CANONICAL MODELLING
An approach for intermediate-level simulation of carbon allocation in
functional-structural plant models
M. RENTON#, D. THORNBY##, J. HANAN###
#
CRC Australian Weed Management, Dept of Agriculture Western Australia
##
Queensland Dept of Primary Industries and Fisheries
###
ARC Centre of Excellence for Integrative Legume Research, ARC Centre for
Complex Systems and Advanced Computational Modelling Centre, University of
Queensland, Brisbane, Australia
Abstract. Functional-structural models that include detailed mechanistic representation of underlying
physiological processes can be difficult and expensive to build, and the resulting models are often very
complicated. This is particularly true when representing carbon allocation, as the various processes
involved are relatively poorly understood. Purely empirical models, on the other hand, are simpler and
easier to construct, but are of limited use in simulating, predicting and explaining the way that plants
adapt and respond to varying environmental conditions. In this chapter, we discuss an intermediate
approach to modelling plant function that can simulate plant responses, including changes in carbon
allocation patterns, without requiring a detailed knowledge of the underlying physiology. In this
approach, plant function is modelled using a ‘canonical’ modelling approach, where processes such as
carbon allocation are represented by a number of fluxes between compartments, and these fluxes are in
turn represented using flux functions of a standard mathematical form. The values of the parameters of
these flux functions are then determined by fitting the global output of the model to global data, rather
than attempting to make the functions represent underlying processes in a quantitatively accurate way.
Here we demonstrate the canonical modelling using an example involving the cotton plant, where two
alternative hypotheses explaining observed compensation after defoliation are represented. We discuss
some potential advantages of this canonical approach over both more mechanistic and more empirical
approaches to representing carbon allocation, and conclude that canonical modelling offers a useful,
flexible and relatively simple way of simulating plant function at an intermediate level of abstraction.
INTRODUCTION
Species-specific variations in the morphogenetic process are key to the diversity of
structure seen in plants, both among species and between individuals of the same
species in different environments. Various aspects of the physiology or function of
151
J. Vos, L.F.M. Marcelis, P.H.B. de Visser, P.C. Struik and J.B. Evers (eds.), FunctionalStructural Plant Modelling in Crop Production, 151-164.
© 2007 Springer. Printed in the Netherlands.
152
M. RENTON ET AL.
the plant are behind these variations, and one of the important differences between
species is the strategy with which a plant allocates available carbon to its different
components.
One way of trying to understand plant morphogenesis is to construct structural
plant models based on the classification and quantification of growth patterns using
analysis of architectural data (Hallé et al. 1978; Hayes et al. 1990; Godin et al. 1999;
Suzuki 2000). Such models, which are designed to describe plant growth without
explicitly representing underlying physiological processes, such as carbon
allocation, can be called descriptive or empirical models. However, such models
cannot capture the interaction between the development of plant structure and
physiological processes such as carbon allocation.
The alternative is to build computational models that represent both the function
and structure of a plant, often termed functional-structural plant models (FSPMs)
(Room et al. 1996; De Reffye et al. 1997; Kurth and Sloboda 1997; Special issue on
functional-structural tree models 1997; Fournier and Andrieu 1998; Special issue
second international workshop on functional-structural tree models 2000; Godin et
al. 2004). Such models can provide a theoretical framework for experimental
investigations aimed at deepening our understanding of plant growth. The general
approach underlying many FS models is to represent the plant as a large number of
interconnected components (such as internodes and leaves). Various physical,
chemical and physiological processes (such as light interception, photosynthesis,
nutrient transport and carbon allocation) that take place within and between these
components are then represented explicitly (Perttunen et al. 1998; Lacointe 2000;
Sievänen et al. 2000; Sinoquet and Le Roux 2000; Le Roux et al. 2001) in what are
called process-based or mechanistic models.
In many situations, we would like to construct a model that does not require indepth experimental investigations or a high level of model complexity, yet is
capable of capturing the most important, interesting or relevant aspects of the
function and structure of the plant. We would need a model that is adaptable and
able to represent causal hypotheses (unlike a purely descriptive or empirical model),
yet is simpler and easier to construct than the detailed process-based models. A
modelling approach that tried to find this balance between the advantages of
empirical and mechanistic modelling could be called an ‘intermediate-level’
approach.
The intermediate-level FS approach discussed here (Renton 2004) is based on
integrating canonical models (Savageau 1969; 1976; Voit 2000) of plant function
with L-system (Lindenmayer 1968a; 1968b; Prusinkiewicz and Lindenmayer 1990)
simulations of plant structural development. Previous presentations of this approach
(Renton et al. 2005a) have focussed on how canonical modelling can be used to
simulate various aspects of plant function in FS models, and how a structural Lsystem model can be linked to an existing canonical model (Kaitaniemi 2000) of
plant function (Renton et al. 2005b). Here we focus on how canonical modelling can
be used to represent carbon allocation in particular.
CANONICAL MODELLING
153
CANONICAL MODELLING OF PLANT FUNCTION
Canonical non-linear models have been employed to model a variety of complex
biological systems (Torres 1996; Voit and Sands 1996a; Martin 1997; Kaitaniemi
2000) in a mathematically standard way (Voit 1991; 2000). The diagrammatic
representation of a canonical model consists of ‘compartments’, ‘fluxes’ and
‘influences’. Figure 1 provides a simple example where compartments are drawn as
circles, fluxes as solid arrows and influences as dashed arrows. Compartments are
associated with a variable that usually represents some real-world quantity. Fluxes
represent flows into, out of, and between compartments. An influence drawn
between a compartment and a flux indicates that the magnitude of the flux depends
on the magnitude of the quantity represented by the compartment variable. It is
generally assumed that a flux is influenced by the compartment that it originates
from, so these influences are not drawn.
Figure 1. Example graphical representation of a canonical model
As a high-level abstraction representing the physiology of a plant, x1 would
represent unallocated or substrate carbon, x2 would represent carbon allocated to the
shoot and x3 would represent carbon allocated to the root. The flux f1 would represent
carbon acquisition through photosynthesis, f2 would represent the allocation of
substrate carbon to the shoot and f3 would represent the allocation of substrate
carbon to the root. The arrow i1 would capture the hypothesis that the rate of carbon
acquisition is affected by leaf biomass and the arrow i2 would represent the
hypothesis that the rate of carbon allocation to the shoot is influenced by the amount
of root biomass. It is also assumed that the allocation of substrate carbon to both
shoot and root will depend on the amount of substrate carbon.
Once this diagrammatic (or compartment) representation of the model has been
formulated, fluxes are then represented as ‘canonical’ (or standardized) functions of
all compartment variables that influence that flux. The standard canonical power-
154
M. RENTON ET AL.
law function consists of a ‘rate constant’ multiplied by the product of all associated
compartment variables, each raised to a constant ‘kinetic order’ power. In Figure 1,
flux f2 originates from the x1 compartment and is influenced by the x3 compartment,
so it would be written as
f 2 ( x1 (t ), x3 (t )) D 2 x1 (t ) k21 x3 (t ) k 23 ,
(1)
where D2 is the rate-constant parameter and k21 and k23 are the kinetic-order
parameters (with subscript pairs indicating the flux being operated on followed by
the compartment-variable identifier).
The way in which these fluxes are aggregated depends on which canonical
formalism is being employed. One of the most commonly used formalisms is a
Generalized Mass Action (GMA) system, where the rate of change of a
compartment variable is written as the sum of all fluxes in, minus all fluxes out.
Using a GMA system in this example means the differential equation for x1 would be
dx1 (t )
dt
f1 ( x2 (t )) f 2 ( x1 (t ), x3 (t )) f 3 ( x1 (t ))
D1 x2 (t )
k12
k 21
D 2 x1 (t ) x3 (t )
k 23
D 3 x1 (t )
(2)
k 31
which is normally written in the abbreviated form
x1
D 1 x 2 k D 2 x1 k x3 k D 3 x1 k
12
21
23
31
(3)
The other two differential equations representing the model would thus be
D 2 x1k x3k
x2
21
x3
D 3 x1k
31
23
(4)
(5)
The procedure for using the canonical power-law modelling approach to
simulate an aspect of plant function can be summarized in five steps, as follows.
First, the important or significant quantities must be identified, and associated with a
compartment and variable. Second, the fluxes or flows into, out of, and between
these compartments must be chosen. Third, the modeller must decide which
quantities affect which of these fluxes, and include these influences in the model. At
this point, we have formulated a compartment model, such as the one shown in
Figure 1, that represents qualitative mechanistic assumptions or hypotheses. In the
fourth step, a canonical form, such as the power-law form, is used to represent each
flux as a function of the influencing variables. Finally, values for the parameters of
CANONICAL MODELLING
155
these flux functions are estimated, using one or more of a number of possible
approaches, including eliminating unnecessary parameters; recognizing constraints
on parameters implied by the modelled system; reformulating equations to give
parameters a clearer significance; scaling variables; using ad hoc manipulation of
parameter values with visual feedback of model output; flux-based estimation and
using computational algorithms to fit model output at the global scale (Voit 2000;
Renton 2004). The parameterized model can be tested and compared against further
data, and modified as necessary.
The first three steps of the canonical modelling process are mechanistic in nature
and, in general, the inclusion of more compartments and connections (fluxes and
influences) corresponds to a more mechanistic model. The last two steps are
empirical in that a general equation form is used to summarize a number of
physiological processes, and parameter values are chosen to ensure that model
output fits the data rather than using an equation form and parameter values
representing a particular mechanism.
LINKING TO A STRUCTURAL REPRESENTATION
A canonical model of plant function can be linked to a representation of plant
structure to create a functional-structural plant model (Renton et al. 2003; Renton
2004; Renton et al. 2005b). The basic strategy in linking a canonical model of
function to a structural L-system model is to make the L-system rules for expansion
(change in size of existing plant components) and/or development (the addition of
new plant components) depend on the state variables in the canonical model. If the
canonical variables correspond to individual structural component characteristics,
such as having a compartment for the size of each leaf, this is a straightforward
process. If the canonical variable represents a global characteristic of the plant, such
as number of components, or plant height or biomass, increases in the variable can
then be ‘shared out’ to create new components and/or to expand the size of existing
components in the L-system model according to hypothesized distribution rules.
These distribution rules may be stochastic, simulating structural variability, and may
also simulate aspects of plant physiology, by taking into account structural and
environmental inputs, such as location within a crown, or availability of light.
CANONICAL MODELLING OF DEFOLIATION IN COTTON
The use of the canonical modelling approach to represent carbon allocation
processes can be demonstrated by considering growth of the cotton plant. The model
is based on a glasshouse experiment that measured cotton plants at regular intervals
as they grew to obtain internode and leaf numbers and lengths, which were
processed to give estimates of leaf and stem biomass for an average plant. Further
details on experimental procedure; data collection and treatment; the results
obtained; and model construction, parameterization and testing are available
elsewhere (Thornby et al. 2003; Renton 2004; Thornby 2004; Renton et al. 2005a).
A basic canonical model of carbon acquisition and allocation, similar to the example
156
M. RENTON ET AL.
given in Section "Canonical modelling of plant function", consists of compartments
for unallocated or substrate resources (xu), for resources fixed in leaf biomass (xl),
and in stem biomass (xs), as shown in Figure 2. The flux fi represents resource
acquisition under the standard conditions in which the plants were grown and fl and
fs represent allocation of resources to leaf and stem, respectively. The flux fo
represents the allocation of resources to all other parts of the plant, or resources that
are used or lost in other ways. The dotted arrow represents the influence of leaf
biomass on the rate of resource acquisition. This diagrammatic
Figure 2. Compartment representation of the basic cotton growth model (top) and the output
of this model compared to the data points (bottom). In this model, xu represents the amount of
unallocated resources, and xl and xs represent the leaf and stem biomass, respectively. The
flux fi represents resource acquisition and fl,, fs and fo represent the allocation of resources to
leaf, stem and other destinations, respectively
representation was translated into a system of canonical equations, which was then
parameterized to fit the data for the control treatment. This basic cotton growth
model was able to simulate the observed data for the control treatment plants, as
shown in Figure 2.
CANONICAL MODELLING
157
To construct the FS model, we began with this basic canonical model and an
existing ‘template’ structural L-system model of cotton development (Hanan and
Hearn 2003; Thornby et al. 2003; Hanan 2004; Thornby 2004). Since we were
interested primarily in defoliation and leaf biomass compensation, the models are
linked using the total leaf biomass variable (xl) of the canonical model. Apart from
this link, the two sub-models run in parallel, with development (that is, the rate of
appearance of new plant parts and the types of parts being produced) being
controlled by the original L-system model independently of the canonical model.
The canonical sub-model is integrated with a very small (approximately continuous)
time step and the structural sub-model is updated with a daily time step. At the end
of each day, the ‘potential growth’ of each immature leaf in the L-system sub-model
is calculated using a function that first rises, then falls with the age of the leaf. The
‘total potential growth’ is calculated as the sum of all biomass requirements for the
potential growth of all immature leaves in the L-system structure; the ‘total actual
growth’ is set to be the amount of new leaf growth indicated by the canonical model
for that day (which generally differs from the total potential growth indicated by the
structural model); and the ‘growth proportion’ is set equal to the ‘total actual
growth’ divided by the ‘total potential growth’. The actual growth of each individual
leaf is then calculated as this ‘growth proportion’ multiplied by the original potential
growth of that leaf. This ensures that the total leaf area in the structural model is the
same as that indicated by the canonical model at the end of the day, and that each
leaf grows according to a sigmoid function in ideal conditions. It also causes the rate
of individual leaf growth to slow when many leaves are growing at the same time.
This model can then be extended to consider more detailed aspects of
physiology. Here, we use the model to investigate different functional hypotheses
regarding physiological responses to defoliation, using data from an experiment
involving defoliation of cotton seedlings (Thornby 2004). In the canonical submodel, defoliation is modelled as a reduction in the leaf compartment value at the
appropriate time. In order for compensation to occur, as was found in the data, there
must be an additional mechanism that increases the leaf production rate following
defoliation. We formulated two alternative explanatory hypotheses regarding this
mechanism. One hypothesis is that the defoliation causes the production of some
signalling compound in the plant, and it is the presence of this compound that causes
increased allocation of resources to leaf production. The other hypothesis is that
leaves constantly produce some compound such that the concentration of this
compound within the plant remains relatively constant unless defoliation occurs, in
which case production, and thus concentration, of this compound drops. According
to this hypothesis, it is this drop in concentration that, in turn, causes the increased
rate of leaf growth. Based on the original cotton growth model (Figure 2), we
158
M. RENTON ET AL.
Figure 3. Compartment representations of canonical models of growth and compensation in
cotton, based on the hypotheses that compensation following defoliation is caused by the
presence (top) or the absence (bottom) of a regulating compound. In these models, xh
represents the amount of the compound in the plant, xc represents the concentration of the
compound, and xb is a Boolean trigger variable that is ‘turned on’ following defoliation for
an amount of time proportional to the level of defoliation. The flux fp represents the
production of the compound and fd represents its degradation
developed the two compartment models shown in Figure 3 to represent these two
alternative hypotheses. These were translated into canonical equations in the
standard way and parameterized to fit the observed data for the two defoliation
treatments used in the experiment. These two alternative models of compensation
were both able to simulate the observed compensation behaviour (Renton 2004;
Figure 4, Renton et al. 2005a).
CANONICAL MODELLING
159
Figure 4. Simulating the effect of 58% defoliation on day 79 using the ‘presence’ model (top)
and the ‘absence’ model (bottom)
The user of the FS model is able to specify if and when a particular leaf is removed
from the structural L-system model; at the same time, the equivalent amount of
defoliation is calculated and simulated in the canonical model. The response to this
defoliation (simulated by the canonical model according to the presence or absence
hypothesis) will involve a boost to leaf biomass allocation in the canonical model,
leading to an increase in the rate at which new leaves will grow in the L-system
160
M. RENTON ET AL.
model. The FS model can thus be used to predict individual leaf biomass (and hence
area), and the change in individual leaf biomass as a result of defoliation. Output
from the FS model is shown in Figure 5.
Figure 5. Example output of the FS model of the cotton plant’s growth and response to
defoliation. The arrow shows where three leaves have been removed in the fourth image.
Structural output, such as leaf sizes over time, can now be checked against data
DISCUSSION
Canonical plant models include a representation of underlying processes,
mechanisms and interactions, and therefore share some of the advantages that
detailed process-based mechanistic models have over empirical models. First, the
process of constructing the model requires the modeller to develop a set of causal
hypotheses regarding the observed behaviour and represent these hypotheses in a
‘formalized’ and precise way. Second, the finished model can have explanatory
value; the model is capable of explaining observed behaviour in terms of underlying
mechanisms. Third, the model can be used to predict behaviour in a wider range of
conditions and situations beyond those in which the original data used to construct
them was collected (Kaitaniemi 2000).
Because of these three attributes, the canonical approach can be used for
exploratory modelling strategies where the model acts as a theoretical framework for
experimental investigations. Because the model is a synthesis of a number of
mechanistic hypotheses regarding the plant’s behaviour, these underlying biological
hypotheses can be falsified or refined by comparing the model output to patterns
observed in experimental data. Versions of the model representing a number of
alternative hypotheses can be constructed, as shown with the ‘absence’ and
‘presence’ versions of the cotton model. The approach allows processes and new
CANONICAL MODELLING
161
variables to be added to the model without changing its overall structure (Voit and
Sands 1996a), which facilitates building these alternative versions. These model
versions can be used to design experiments or suggest field observations that will
distinguish between these different versions, and thus help determine the validity of
the alternative hypotheses. In this way, an ongoing process of model refinement and
data collection will lead to an increased understanding of biological mechanisms.
However, unlike the detailed process-based approach, the canonical modelling
approach can be used when detailed hypotheses or data regarding underlying
physiological or chemical processes is not available, such as with carbon allocation.
This is because the compartment model can be formulated at a relatively abstract
level, the flux functions are of a standard power-law form, and the parameters of
these functions can be found empirically by fitting model output to relatively easily
observed global variables, rather than by direct measurement of physiological
processes (Voit and Sands 1996a; 1996b; Kaitaniemi 2000). For example, the cotton
model is based on hypotheses regarding causal mechanisms and physiological
interactions, but these were formulated and represented at a general and nonquantitative level. In a situation such as this, where there seems to be little
conclusive evidence for what physiological processes are involved in changes to
carbon allocation patterns after defoliation (Thornby 2004), a more abstract style of
modelling, such as the canonical modelling approach, may be the only option.
Even if the eventual aim is to construct a more detailed and process-based
model, a canonical model can act as a ‘placeholder’ for modelling parts of the
system that are not yet well understood. More realistic modelling can be used to
represent those parts for which a more detailed understanding already exists. For
example, a detailed process-based model of photosynthesis could replace the
‘resource acquisition’ flux fi in the canonical cotton model, while carbon allocation
would remain represented by canonical flux functions. In this way, the model can
contain both abstract and more explicit representations of plant function.
Due to the flexibility of the canonical form of their flux functions, canonical
models will tend to fit data more accurately than most empirical models. When
parameterized to fit global data, they will also be more accurate than process-based
models with parameters fitted at the level of the underlying processes (Renton 2004;
Renton et al. 2005a). Canonical models can be used directly, or adapted, to simulate
a range of aspects of plant function, such as resource acquisition and growth, limits
on growth, storage, allocation and suppression, as well as non-continuous changes in
behaviour, such as the triggering of fruiting (Renton 2004; Renton et al. 2005a).
They can also be used to model environmental influence on plant growth (Voit 1993;
Renton 2004); prioritized allocation of resources (Renton 2004); conversion
between continuous biomass quantities and discrete numbers of shoots at the end of
a season (Kaitaniemi 2000); and thinning dynamics (Voit 1988) and biomass
budgets and growth (Voit and Sands 1996a; 1996b) in tree stands. These processes
can be represented canonically at whatever scale is most relevant: whole fields,
individual plants, or plant components.
The usefulness of a canonical model of plant function depends on the level of
abstraction of the model. Very descriptive canonical models (Renton et al. 2005a)
can be used for visualizing, describing or communicating experimental results. A
162
M. RENTON ET AL.
more mechanistic canonical model, such as the cotton example, can be used as a
theoretical framework in an ongoing process of experimental investigation, as
discussed above. While detailed process-based models tend to aim for a high degree
of realism, and thus risk becoming very complex, the canonical approach helps build
a functional model that is ‘just complex enough’, by including only the relevant
processes at the necessary level of detail for a particular problem. If the goal is to
produce a very realistic and detailed model for research purposes, then the canonical
approach may not be appropriate. Nevertheless, if the testing and refining of a
canonical model were continued against a broad range of experimental results, the
model may eventually have the potential to be used for management applications,
such as decision-making, prediction or control.
Canonical modelling of plant function can certainly be employed without a
structural representation, but adding structure increases the usefulness of the model
in different ways, depending on the model’s level of abstraction. Adding a
corresponding representation of plant structure enhances the ways that a descriptive
model can be used for visualizing, describing or communicating information. With
the addition of structure, a descriptive canonical model could also act as a sub-model
within broader theoretical investigations, such as looking at how light interception
patterns change with the growth of the plant, or understanding patterns of insect
movement or spore dispersal through the structure of the plant. Adding structure to a
more mechanistic canonical model, like the cotton model, creates many extra
aspects of model output, such as branching patterns, size of individual components,
and even the visual appearance of the plant, that can be checked against
experimental data. Furthermore, adding a model of structure that interacts with the
canonical model of physiology in some way – through simulations of endogenous
signalling or light capture by individual leaves (Renton et al. 2005b), for example –
can enhance the level of mechanism of the model.
The canonical modelling approach provides a means of modelling plant
processes, such as carbon allocation, at an ‘intermediate’ level of abstraction,
between that of detailed process-based mechanistic models aiming for a high degree
of realism, and empirical models aiming simply to describe observed patterns. The
resulting functional model can then be linked to structural plant models to produce
FS plant models. Future research could further develop the approach’s usefulness for
plant modelling by investigating how the approach can be applied to a wider range
of experimental and modelling situations; comparing the benefits of different
canonical function forms; simplifying and possibly automating the process of
parameter estimation; and exploring in greater depth and eventually formalizing the
process of linking the canonical model to the structural model.
REFERENCES
De Reffye, P., Fourcaud, T., Blaise, F., et al., 1997. A functional model of tree growth and tree
architecture. Silva Fennica, 31 (3), 297-311.
Fournier, C. and Andrieu, B., 1998. A 3D architectural and process-based model of maize development.
Annals of Botany, 81 (2), 233-250.
Godin, C., Guedon, Y. and Costes, E., 1999. Exploration of a plant architecture database with the
AMAPmod software illustrated on an apple tree hybrid family. Agronomie, 19 (3/4), 163-184.
CANONICAL MODELLING
163
Godin, C., Hanan, J., Kurth, W., et al., 2004. Proceedings of the 4th International workshop on
functional-structural plant models: abstracts of papers and posters, 07-11 June 2004. UMR AMAP,
Montpellier. [http://amap.cirad.fr/workshop/FSPM04/proceedings/4thFSPM04Proceddings.pdf]
Hallé, F., Oldeman, R.A.A. and Tomlinson, P.B., 1978. Tropical trees and forests: an architectural
analysis. Springer-Verlag, Berlin.
Hanan, J., 2004. Modelling cotton (Gossypium hirsutum L.) with L-systems: A template model for
incorporating physiology. In: Godin, C., Hanan, J., Kurth, W., et al. eds. Proceedings of the 4th
International workshop on functional-structural plant models: abstracts of papers and posters, 07-11
June 2004. UMR AMAP, Montpellier, 268-272. [http://amap.cirad.fr/workshop/FSPM04/
proceedings/4thFSPM04Proceddings.pdf]
Hanan, J.S. and Hearn, A.B., 2003. Linking physiological and architectural models of cotton. Agricultural
Systems, 75 (1), 47-77.
Hayes, P.A., Steeves, T.A. and Neal, B.R., 1990. An architectural analysis of Shepherdia canadensis and
Shepherdia argentea (Elaeagnaceae): the architectural models. Canadian Journal of Botany, 68 (4),
719-725.
Kaitaniemi, P., 2000. A canonical model of tree resource allocation after defoliation and bud
consumption. Ecological Modelling, 129 (2/3), 259-272.
Kurth, W. and Sloboda, B., 1997. Growth grammars simulating trees: an extension of L-systems
incorporating local variables and sensitivity. Silva Fennica, 31 (3), 285-295.
Lacointe, A., 2000. Carbon allocation among tree organs: a review of basic processes and representation
in functional-structural tree models. Annals of Forest Science, 57 (5/6), 521-533.
Le Roux, X., Lacointe, A., Escobar-Gutierrez, A., et al., 2001. Carbon-based models of individual tree
growth: a critical appraisal. Annals of Forest Science, 58 (5), 469-506.
Lindenmayer, A., 1968a. Mathematical models for cellular interaction in development. Part 1: Filaments
with one-sided inputs. Journal of Theoretical Biology, 18 (3), 280-299.
Lindenmayer, A., 1968b. Mathematical models for cellular interactions in development. Part 2: Simple
and branching filaments with two-sided inputs. Journal of Theoretical Biology, 18 (3), 300-315.
Martin, P.G., 1997. The use of canonical S-system modelling for condensation of complex dynamic
models. Ecological Modelling, 103 (1), 43-70.
Perttunen, J., Sievänen, R. and Nikinmaa, E., 1998. LIGNUM: a model combining the structure and the
functioning of trees. Ecological Modelling, 108 (1/3), 189-198.
Prusinkiewicz, P. and Lindenmayer, A., 1990. The algorithmic beauty of plants. Springer-Verlag, New
York.
Renton, M., 2004. Function, form and frangipanis: modelling the patterns of plant growth. University of
Queensland, Brisbane. Ph.D. Thesis University of Queensland
Renton, M., Hanan, J. and Burrage, K., 2005a. Using the canonical modelling approach to simplify the
simulation of function in functional-structural plant models. New Phytologist, 166 (3), 845-857.
Renton, M., Hanan, J. and Kaitaniemi, P., 2003. The inside story: including physiology in structural plant
models. In: Proceedings of the International Conference on Computer Graphics and Interactive
Techniques in Australasia and South-East Asia (GRAPHITE2003), Melbourne, Australia, 11-14
February 2003. ACM Press, New York, 95-102.
Renton, M., Kaitaniemi, P. and Hanan, J., 2005b. Functional-structural plant modelling using a
combination of architectural analysis, L-systems and a canonical model of function. Ecological
Modelling, 184 (2/4), 277-298.
Room, P., Hanan, J.S. and Prusinkiewicz, P., 1996. Virtual plants: new perspectives for ecologists,
pathologists and agricultural scientists. Trends in Plant Science, 1 (1), 33-38.
Savageau, M.A., 1969. Biochemical systems analysis. I. Some mathematical properties of the rate law for
the component enzymatic reactions. Journal of Theoretical Biology, 25 (3), 365-369.
Savageau, M.A., 1976. Biochemical systems analysis: a study of function and design in molecular
biology. Addison-Wesley, Reading.
Sievänen, R., Nikinmaa, E., Nygren, P., et al., 2000. Components of functional-structural tree models.
Annals of Forest Science, 57, 399-412.
Sinoquet, H. and Le Roux, X., 2000. Short term interactions between tree foliage and the aerial
environment: an overview of modelling approaches available for tree structure-function models.
Annals of Forest Science, 57 (5/6), 477-496.
Special issue on functional-structural tree models, 1997. Silva Fennica, 31 (3).
164
M. RENTON ET AL.
Special issue second international workshop on functional-structural tree models, 2000. Annals of Forest
Science, 57 (5/6).
Suzuki, A., 2000. Patterns of vegetative growth and reproduction in relation to branch orders: the plant as
a spatially structured population. Trees: Structure and Function, 14 (6), 329-333.
Thornby, D., 2004. Using computational tools to investigate the responses of cotton plants (Gossypium
hirsutum L.) to defoliation. University of Queensland, Brisbane. PhD thesis University of
Queensland
Thornby, D., Renton, M. and Hanan, J., 2003. Using computational plant science tools to investigate
morphological aspects of compensatory growth. Lecture notes in computer science, 2660, 708-717.
Torres, N.V., 1996. S-system modelling approach to ecosystem: application to a study of magnesium
flow in a tropical forest. Ecological Modelling, 89 (1/3), 109-120.
Voit, E.O., 1988. Dynamics of self-thinning plant stands. Annals of Botany, 62 (1), 67-78.
Voit, E.O., 1991. Canonical nonlinear modeling: s-system approach to understanding complexity. Van
Nostrand Reinhold, New York.
Voit, E.O., 1993. S-system modeling of complex systems with chaotic input. Environmetrics, 4 (2), 153186.
Voit, E.O., 2000. Computational analysis of biochemical systems: a practical guide for biochemists and
molecular biologists. Cambridge University Press, Cambridge.
Voit, E.O. and Sands, P.J., 1996a. Modeling forest growth. I. Canonical approach. Ecological Modelling,
86 (1), 51-71.
Voit, E.O. and Sands, P.J., 1996b. Modeling forest growth. II. Biomass partitioning in Scots pine.
Ecological Modelling, 86 (1), 73-89.